The Reflective Educator

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Year: 2013 (page 1 of 15)

Fake World Versus Real World

Like many math teachers, I have been following Dan Meyer’s discussion on “fake world” math tasks versus “real world” math tasks with interest, especially since one of my early blog posts was on this subject and one of the presentations I do for teachers is on this very topic.

My observation is that it is not the task that defines whether it is fake or real, it is the person doing the taskOur work then should focus on developing criteria on what makes tasks real for children, and then see which tasks support the criteria we establish. Here are some criteria I think we should consider when developing tasks for students, aside from the obvious; the task should engage students in mathematical thinking.
 

  • Relationships:

    Children do mathematics in a socio-emotional context, and virtually always with the support of a mentor (peer, parent, teacher, etc…). A strong relationship with someone who can support their mathematical reasoning is critical. Tasks which the mentor finds interesting or appealing are more likely to be interesting or appealing to the children they support. There are lots of stories of lone mathematicians working in secret for years on developing mathematics, but I do not know of any stories of children doing the same.
     

  • Questions:  

    Children are more engaged with tasks that they have questions about. By this, I do not mean the pretend inquiry questions that people sometimes start a unit with (Imagine here a teacher-led discussion that leads to a wall full of questions children made up on the spot to satisfy the “let’s make up our inquiry questions” game…), but actual questions that students have about the world and the objects in it.
     

  • Access:  

    Children need to be able to access the task and to do it substantially by themselves. A task where the adult with them has to do most of the work, either physical or otherwise, quickly becomes much less interesting for children. I built (from a kit) a compressed air rocket a couple of months ago with my son. He spent most of the time bored as I fit pieces together and he occasionally got to glue things together. Although he was very interested in the final product (who wouldn’t be interested in something that can shoot paper rockets up 50 metres in the air?), the process of making the rocket was tedious because he only had periphery access to the building process. This is true of mathematical tasks as well. Tasks where children have to rely on a list of “how to” steps provided by someone else are rarely interesting, unless some significant thinking has to occur to make the steps useful.
     

  • Challenge:  

    Children often like to do things because they are challenging so we have to be careful not to make things we ask children to do to be too simple when we are ensuring they have access. I once gave the Seven Bridges of Königsberg problem to my 9th grade class. The problem was accessible because every student felt like they have a possible solution path (ie. draw a picture), but it was challenging so the students kept working on it. Working on the problem became infectious, and soon, most of the 9th grade math classes in the school worked on the problem at least a little. Some of my 9th graders spent three weeks trying to solve the problem before finally coming to ask me to prove that it was impossible.
     

  • Familiarity:  

    Children do not have questions about things with which they lack familiarity. If the context you are using is completely unfamiliar to children, then they aren’t going to have questions. If one takes the time to develop context around a situation (ie. story-telling), then it is more likely that children will begin to wonder about it. Every good game as a plot that hooks the player into the game, a good math task should do the same thing.

 

Note that these criteria all lead to an important conclusion; some tasks will be considered fake by some students, and real by others. It is important to note too that because of our shared society and context, there are some tasks which will be real for almost all students, and there are other tasks which will be fake for almost all students.

[Global Math] Recording of my presentation on Teacher and Student questions

I just noticed that I can embed recordings of presentations from Big Marker, so I decided to embed my presentation for the Global Math department here. The presentation topic was “Teacher and Student Questions.” You can view the slides here.

 

My son, the math teacher

My son went to a full day session to meet up with some other children around his age who are being homeschooled. During this time he had a class on engineering with Lego, and a lot of free time with which to play and socialize with the other children.

Interestingly enough, the kids decide to play the Game of School. They found a chalkboard and took turns pretending to be the teacher and explaining things to the other students. My son decided to be the math teacher, and according to him, this is what he tried to explain.

“9 x 10 = 90”  and “90 x 10 = ?” and “? x 10 = a different ?” and “a different ? x 10 = another ?” and so on.

In other words, he was trying to show a pattern with multiplication by 10, and he decided to use a place-holder to show that the result of his multiplication from the first times 10 would be used in the next calculation.

We have another word we use commonly in mathematics for these place-holders. We call them variables. My son is 7. He has had no formal instruction on variables, and I have certainly never talked about the idea in our discussions about math. This is him inventing a new-to-him mathematical idea to help describe a process to another child.

Why is my son using variables in this explanation? I think it’s because he had a need for them. Does he understand the concept of variable completely? I doubt it, but this is a good start.

My child certainly has a stronger-than-usual background in mathematics (he has me as his father, and we talk about math ideas and numbers a lot) but if he can invent this concept, other children can too, given the necessary experiences.

 

 

Workshop on Social Media for Students

I recently facilitated a workshop on social media for a school. I created two short videos to act as discussion starters, and then I created a workshop structure around the video clips that the school could use to facilitate the discussion with their students.

Outline of a workshop for students on social media:

  1. Introduce me (Optional):

    David Wees works with New Visions for Public Schools in the a2i project. He spent the last four years working as a educational technology expert for a school in Vancouver, Canada. One interesting fact about David; he has lived and worked in four different countries. He is going to share some information he has learned about social media with us today, with video two short videos he created.

     

  2. Show the clip titled: “Social Media Part 1
    (Optional: Start with students sharing out what they know about the Internet and social media first)

     

  3. Discussion (have students work in groups of 3 or 4 students) – 5 minutes or so

    Some possible questions:

    Print the following questions for students to talk about (add more questions as you see fit)

    What kinds of things have I posted online?
    Who can access these things?
    What can I do to limit access to things I post online?
    What could happen to me if I post things online?

     

  4. Whole group discussion (5 minutes or so)

    Have different groups share out points from their discussion.

     

  5. Show the clip titled: “Social Media Part 2

     

  6. Discussion (have students work in groups of 3 or 4 students) – 5 minutes or so

    What is in my digital footprint?
    What can I do to build a stronger, more positive digital footprint for the future?
    What kinds of things should I avoid doing?
    What kinds of things should I make sure I do?

     

  7. Whole group discussion (5 minutes or so)

    Some possible questions:
    (Recommended: Make a chart for the whole room to see of the positive and negative things about sharing stuff online)

    What are some positive things we can share?
    What are the some of the negative things we should avoid sharing?
    What can we do if we find out someone else is sharing negative stuff about us?
    What is the relationship between what we share, and what people think of us?

     

  8. Exit slip:

    Have students write down three things they learned from the workshop today. Read what they wrote and use it to inform a future follow-up session on social media.

 

*Note that I have very much simplified the “how the Internet works” portion of the video as my aim is to get kids talking about it rather than knowing all of the technical details.

Learning about shape

Picture of my son placing a shape through a hole

 

As I watched my son over the past few days learn about shape, I am struck by not only how much we need to learn to make sense of the world, but also by how even simple things cannot be taken granted as known by children.

My youngest son, who is about 20 months old now, is learning how to take a small piece of yellow plastic that forms a shape (an oval, a circle, a hexagon, and 4 other shapes) and push it through a hole that is only slightly larger than the shape he is pushing through it. The task requires him to pay close attention to both the shape he has in his hand, the orientation of that shape, and the hole he is trying to force it through.

At first he pretty much chooses shapes at random, and tries to jam into the hole. Eventually he finds a match and gets a shape through the hole, or he gives up. He rarely turns the shape in his hand, or the container the shape goes into.

Eventually he learns that he has to often rotate the shape in his hand, and so he picks up a random shape and tries to shove it in a random hole, and when it doesn’t work the first time, he says, “No…no…no,” and rotates the shape a bit to try again.

Soon he has made a few matches, and is always able to easily find the right hole for the circular shape and the oval shape, and soon after the plus sign shape. For all of the other shapes, he continues to try to randomly match shapes to holes. He looks at the shape to identify it as one of the ones he knows, but he does not seem to connect the general idea that the shape he sees in the piece should match the shape he sees in the hole.

The next day, I am surprised to discover that he can match almost all of the shapes, and frequently looks at the shape to see which one he has, and then which hole he should put the shape in. He often rotates the shape several times when he is sure he has the right hole. When he is not sure, he tends to give up quickly and go back to a shape with which he is very familiar.

In a few days, he goes from basically using a random matching strategy to carefully looking at the shape in order to be able to match it. At this stage the only shapes that stump him regularly are the regular pentagon, and the regular hexagon. 

I notice however that his ability to do other puzzles does not seem noticeably improved. It’s like his learning is restricted to this one very narrow context, and within this narrow context, he has either just learned to match each of the individual patterns or possibly he recognizes a small generalization; for this puzzle the shape should match the hole. As the months progress, I will continue to watch how his understanding of shape grows and develops.

It is fascinating to me to see that shape is a learned concept and that even what seem like simple generalizations are learned. It makes me wonder what concepts my students may not have fully developed, even by the time they arrived to me in high school.

A conversation with my son on place value

This is an excerpt from a conversation I had with my son while we were walking from the subway to the theatre.

My son: Daddy, let’s play a number game.
Me: Okay. What’s seven billion, nine hundred and ninety-nine million, nine hundred and ninety-nine plus one?
My son: That’s too big Daddy, I can’t add those!!
Me: Okay, let’s try a simpler problem. What’s nine plus one?
Son: Ten.
Me: Ninety-nine plus one?
Son: One hundred.
Me: Nine hundred and ninety-nine plus one?
Son: One thousand.
Me: Nine thousand, nine hundred and ninety-nine plus one?
Son: I don’t know how to say the next number. Oh wait! TEN thousand (proudly).
Me: Ninety-nine thousand, nine hundred and ninety-nine plus one?
Son: One hundred thousand.
Me: Nine hundred and ninety-nine thousand, nine hundred and ninety-nine plus one?
Son: How do you say one thousand thousands?
Me: One million.
Son (laughs): Okay the last one is one million.
Me (continuing): What’s nine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nineplus one?
Son: Ten million.
Me: What’s ninety-nine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nine plus one?
Son: One hundred million.
Me: Nine hundred and ninety-nine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nine plus one?
Son: How do you say one thousand millions?
Me: One billion.
Son: That’s the answer then, one billion.
Me: Okay, now try the first problem. What’s seven billion, nine hundred and ninety-nine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nine plus one?
Son (no hesitation): Eight billion.
Son: What’s one thousand billions?
Me: One trillion.
Son: What’s one thousand trillions?
Me: One quadrillion.
Son (giggles): And then?
Me: Quintillion
Son: What’s next!
Me: Sextillions, then Septillions, then Octillions, then Nonillions, then probably Decillions.
Son: What’s next?
Me: Probably Endecillions1 and Dodecillions1, but that’s the limit of my Greek.
Me: What if we played our adding game forever?
Son: Infinity! But we’d have to play in Heaven because even if we played until the end of our lives, we still wouldn’t reach infinity.
(Leads to a long discussion on whether heaven exists and where we go when we die.)

This kind of conversation, between my son and I, is typical as we have a lot of conversations about numbers. In this case, I presented him with a challenging problem, and he was not able to do it. I then used George Pólya’s “trick” of asking my son simpler problems which led up to him seeing how to solve the more complicated problem. Does this mean that my son understands place value, or even all the numbers he was able to say? Probably not, our conversation was entirely linguistic, but it’s a start.

 

1.Here is a list of the names of the large numbers. Notice that my two guesses are actually wrong (but close!).

 

 

Teaching proof

I’m currently working on creating a sample sequence of lessons for teachers to use for a geometry unit. At this stage, students will have been exposed to (but will not necessarily have learned all of) geometric transformations, constructions, and some review on geometric vocabulary.

My objective is to create a sequence of lessons which include:

  • embedded formative assessment,
  • opportunities for students to discuss student thinking,
  • opportunities for students to make sense of the idea of proof,
  • opportunities for students to prove geometric proofs,
  • opportunities for students to engage in mathematical inquiry,
  • engaging tasks that help get kids excited about mathematical arguments.

The last time I taught geometry, I think I maybe managed to to hit 1 or 2 points from this list, so I have given myself a tall challenge. I need help.

One of my university professors once told me, “Something is proven true when everyone stops arguing that it isn’t” so with that in mind, here is a vague idea that I am considering and need help fleshing out:

cycle of proof: write proposition => construct argument => share argument => debug argument

Basically, students would spend their time during this unit making interesting geometric observations and then attempting to prove to each other that these arguments are true. Part of their time would be exploring geometric objects, possibly through constructions and possibly through looking at (interactive?) diagrams. During this time, when they see something that they think might be true, they create a proof (in everyday language that they understand) that it is true, and then they present their argument, either in small groups or to the whole class.

Here is an example from my imagination. Suppose students have access to this online geometric construction tool (or lots of paper, a compass, and a straight edge) and when playing around, one of them creates this construction.

construction of a hexagon
 

Student 1: “Oh wow, that’s pretty cool. I made a regular hexagon.”
Student 2: “It sure looks like a hexagon, but remember what Mr. Wees said? It might be just really close to a regular hexagon. How do we know for sure it is one?”
Student 1: “Hrmm. A regular hexagon has all the sides the same length, and this looks like all the sides are the same length, so it must be a hexagon.”
Student 3: “How do you know for sure the sides are all the same length?”
Student 4: “Yah, maybe they are like, one or two pixels off or something.”
Student 2: “How did you draw this? Can you show me how?”
Student 1 shows her group members how she came up with the construction. “See, it’s a regular hexagon.”
Student 3: “I notice that when you made each of the sides you used a circle.”
Student 4: “Yeah, I noticed that too.”
Student 2: “And all of the circles you used were the same size.”
Student 4: “Well, not all of them. There are bigger and smaller circles.”
Student 2: “Okay well all of the smaller circles are the same size.”
Student 1: “How do you know that?”
Student 2: “They all have the same radius, see? These two circles have the same radius, and these two, and these two. They are all connected so they all have the same radius.”
Student 4: “Ooooh, I have an idea…”
Student 1: “Me too! Those radiuses that he pointed out are the same as the lengths of the sides of the hexagon so if they are all the same, then all of the sides of the hexagon are the same size. Done!”
Student 3: “Is it possible to make a hexagon where all the sides are the same length, but the hexagon is still not a regular hexagon?”

 

Obviously this is an idealized situation, and maybe a bit unrealistic but this is where I would like students to end up. What kinds of classroom conditions would lead to students being able to do this? How would the class be structured? What kinds of supports would a teacher have to give to help support students?

Why teach math?

Why do we teach math?

Mathematical procedures
(Image source)

It could be because the mathematical procedures that are taught in schools will be useful to students later, but I am pretty sure this is false. Almost everyone forgets those procedures as they get older because most people in our society use virtually none of the procedures they learned in school in their day-to-day life. Obviously there are engineers, mathematicians, and scientists who use the mathematics they have learned, possibly on a daily basis, but I think if you dig deeper into the work they do, many of these people use tools to help to do their work (like Mathematica, for example), look up the finer details of mathematical procedures that they do not use often, or who use only a very specialized portion of their mathematical knowledge regularly.

It could be that we want to expose students to different ways of thinking about the world. In this case we would be less concerned with the exact set of mathematical procedures they have learned, and more concerned with learning mathematics as a way of thinking and knowing. I see little evidence that this is an explicit goal of mathematics instruction given that; the students are assessed only on the procedures, teachers are assessed on their students understandings of those procedures, and that the set of mathematical procedures we want students to know is so prescribed such that it is virtually identical around the world.

It could be that we would like students to learn transferable problem solving skills. In this case, we want to teach mathematics in such a way as to promote the likelihood that students will be able to transfer what they learn to other areas. Cross-disciplinary study would be the norm, rather than the exception. It turns out that “teaching skills that transfer” is not as simple as one thinks. In fact, my understanding is that most of the times when people learn skills in one context, they do not end up transfering those skills to other contexts. Instruction that aims for transferable skills has to provide opportunities for students to make connections between different areas, reflect on what they have learned, and develop metacognitive strategies so that students think about their thinking. What evidence is there that these types of activities are a regular part of math classes?

Mandelbrot set

It could be that we would like students to see the beauty and elegance of mathematics. One way to do this could be through exploring mathematical art. Another might be to look at some famous examples of truly elegant uses of mathematics. We could also ask students to talk about mathematics in the abstract and come to a shared understanding of what elegance and beauty in mathematics mean. As far as I know, none of these activities is a common one in math classes. It is depressing to me that this way of thinking which has so much beauty in it is shared in such a way that almost no one in our society ever gets to experience beautiful mathematics.

If one or more of the reasons I suggested above is something you think is a good reason to teach mathematics, how are you ensuring that you meet this goal with what happens for students in your classroom? 

What other reasons are there to teach mathematics?

 

How do you define variable?

I recently read the Common Core standards for Math for grade 6, which is where the concept of variable appears to be introduced. The standard in question reads:

Write, read, and evaluate expressions in which letters stand for numbers.

I tweeted this out, and one person responded with this observation:

Clearly this definition of variable is limited, and possibly misleading.

 

How do you define variable with your students? How do you introduce the topic?

 

 

Getting started with programming JavaScript

JavaScript is one of the primary web programming languages. It is the language used in Google Apps Scripting, and in HTML5 interactive documents. Chances are pretty good that most of the interactive features that you see in websites were coded in JavaScript.

Much of the power of JavaScript comes from its power to manipulate and modify HTML, which is the markup language that websites are written in. Therefore, if you want to learn JavaScript, you will sooner or later need to learn HTML (and CSS) as well.

You will want these things to get started:

  1. At least one web browser. This will be where you run your code. I recommend using Firefox with the extension Firebug installed. Firebug is incredibly useful for finding bugs in your code, and for debugging. Google Chrome is also useful for some simple debugging as well. Note: JavaScript does not always run the same in every browser, so one reason to have multiple browsers is to test that the code you write works in all of them.
  2. A good text editor. I recommend Programmer’s Notepad for a PC, and TextWrangler for Mac, both of which are free.
  3. A language reference. I personally like Mozilla’s reference here
  4. Some understanding of how web technologies work. This reference guide, again by Mozilla, is useful.
  5. A folder on your computer dedicated to your projects. It is a good idea to be organized from the beginning.
  6. Blocks of time in your schedule to spend working on programming. Programming is something that will take concentration and focus, and is hard to learn in 10 minute intervals.
  7. I also recommend buying a reference/tutorial book. I found JavaScript for Dummies helpful as a starting place, for example.
  8. A JavaScript library that professionals use (I recommend jQuery). This will allow you to create more interesting projects, while at the same time helping you develop an understanding of JavaScript ideas.
  9. An understanding of the design cycle is useful.

Once you have the things above, my recommendation is try to duplicate someone else’s project. This will help you ensure that you have your development process set up correctly on your computer. Your JavaScript files are loaded by the HTML page you create, and so end up linked in it. Here is a sample template you can use for your JavaScript development. There are a lot of ideas that will want to unpack if you use this template.

Simple HTML template

My workflow for programming in JavaScript is to have the text files I am editing open in one program, and the output open in a browser. When I make changes to my code, and want to see the output, I simply save the changes, and then refresh the page that has my JavaScript in it.

In a future post, I will share a screen-cast of starting my own JavaScript project, and talk through the creation of this project.